That sounds like a netting device to place over a bed to keep out mosquitoes and other insects. This was standard equipment, especially for non-natives, for sleeping in the tropics but, no doubt, used in temperate climates as well. There are those of us who can remember baby carriages in temperate climate areas having a mosquito netting accessory so it seems reasonable that some persons would protect themselves and their children when in bed. In connection with this it should be noted that not everyone in New York had (has?) screening on their windows and certainly kept the windows open during the warm months/mosquito season.